We use mathematical models to study the mechanisms of oscillatory electrical activity arising from ion channels in cell membranes and modulated by intracellular chemical processes. We are interested in both the behavior of single cells and the ways in which cells communicate and modify each other's behavior. Our main application has been to the biophysical basis of insulin secretion in pancreatic beta-cells. We have examined bursting oscillations in membrane potential and the role of electrical coupling between cells in the islet of Langerhans. Long term goals are to understand how the membrane dynamics interact with intracellular events to regulate secretion. We also compare, contrast, and generalize to other secretory cells and neurons, including GnRH-secreting hypothalamic neurons, pituitary somatotrophs, and fast neurotransmitter secretion at nerve terminals.[unreadable] [unreadable] Our primary tool is the numerical solution of ordinary and partial differential equations. We use analytical, geometrical, graphical, and numerical techniques from the mathematical theory of dynamical systems to help construct and interpret the models. Perturbation techniques are used to get approximate analytical results in special cases. We study both detailed biophysical models and simplified models which are more amenable to analysis. Such an approach aids in the isolation of the essential or minimal mechanisms underlying phenomena, the search for general principles, and the application of concepts and analogies from other fields. Another role for our group is to act as a bridge between the mathematical and biological disciplines. This includes disseminating the insights of mathematical work to biologists in accessible language and alerting mathematicians and other theoreticians to new and challenging problems arising from biological issues.[unreadable] [unreadable] Recent work on this project includes: [unreadable] [unreadable] 1. (Combined Electrical and Metabolic Oscillations in Pancreatic Islets) We have applied our previously proposed model for calcium oscillations driven by the interaction of calcium feedback onto ion channels and glycolytic oscillations that drive K(ATP) conductance (Bertram et al, 2004) to systematically simulate data on the response of islets to glucose. We find that the model can account for the diverse responses of all three classes of oscillations (fast, slow and compound). We find in particular, in both the experiments and the simulations, that slow islets respond to an increase in glucose similarly to fast islets, with an increase in the duration of the active phases relative to the silent phases. This contrasts to previous claims that slow oscillations are insensitive to glucose, which were based largely on the behavior of isolated cells rather than islets. In most cases, the period and amplitude of the oscillations were not much affected by increases in glucose, as others have seen, but in a significant minority of case we did observe a dramatic increase in both period and amplitude, which we attribute to a change from ionic oscillations to metabolic oscillations. This observation is important because it shows that both fast, presumptively ionic, and slow, presumptively glycolytic, oscillations can occur in the same islet, indicating that these two modes are both part of the natural islet repertoire and not an artifact of the preparation. Finally, we were able to predict with the model that it should be possible to see slow, presumptively glycolytic oscillations, with only small calcium oscillations that are large enough to serve as a reporter for the metabolic oscillations but are likely below threshold to activate calcium-dependent ion channels or other processes. This was confirmed experimentally by both lowering glucose and by inhibiting glucose metabolism pharmacologically. This demonstrates that the slow oscillations can occur in the absence of calcium feedback, in contrast to other models in which calcium feedback is critical for slow oscillations (Nunemaker et al, 2006). [unreadable] [unreadable] Future investigations will focus on demonstrating that the slow oscillations are in fact metabolic in origin. In order to facilitate this, we have extended the model to include a more complete representation of mitochondrial respiration, particularly to incorporate the experimentally measurable variables of oxygen consumption, NADH, and mitochondrial membrane potential (Bertram et al, in press). The extended model has allowed us to offer plausible explanations for why raising calcium increases NADH when glucose is low but lowers NADH when glucose is high: the glucose level changes the balance between the activating effect of calcium on citric acid cycle dehydrogenases and the inhibiting effect of mitochondrial calcium uptake to depolarize mitochondrial membrane potential. The model also shows that blocking calcium entry with the K(ATP) channel opener diazoxide can terminate glycolytic oscillations by reducing the demand for ATP by calcium pumps and thereby switching off the key glycolytic enzyme, phosphofructokinase. This accounts for data showing the diazoxide can stop slow oscillations and rebuts a potential objection to the model.[unreadable] [unreadable] We have used the combined glycolytic-ionic model to address further how beta-cells synchronize within islets. We have found that diffusion of calcium or the glycolytic intermediate glucose-6-phosphate between beta-cells can kill the oscillations rather than enhancing synchrony. We have studied this mathematically and shown that this is due to a pitchfork bifurcation, in which the cells go to disparate steady states, some with high activity and some with low activity. This behavior coexists with the synchronized oscillatory behavior. As reported previously, we feel that this may account for the observation that over-expression of gap junction proteins can convert slow, presumptively glycolytic, calcium oscillations into fast, presumptively ionic, oscillations. This is because the bifurcation kills the slow oscillation leaving the fast oscillation intact. In addition, it offers a new explanation for our previously reported experimental observation that removal and readdition of glucose can convert slow islets to fast ones (Tsaneva-Atanasova et al, 2006).[unreadable] [unreadable] 3. (Synaptic Transmission) We previously published several papers describing a model for fast synaptic facilitation (an increase in synaptic neurotransmitter release during trains of stimulus pulses with interpulse intervals of tens to hundreds of msec) based on slow unbinding of calcium from one or more binding sites. The slow unbinding confers short-term memory because some of the calcium that binds during a previous pulse remains bound when a succeeding stimulus arrives, increasing the probability of release. One objection that has been raised to this model is based on data showing that calcium buffers reduce facilitatoin, which has been taken as an argument in favor of an alternative hypothesis that faciliation is due to accumulation of free, rather than bound, calcium. We have now developed a model which takes into account both free and bound calcium and shown that the data are not incompatible with our core idea that accumulation of bound calcium makes the major contribution to facilitation. Indeed, the bound calcium model is simpler and performs better than the other leading models proposed to date (Matveev et al, 2006).